Take a representation , and pick a basis of and the dual basis of . We define the map by . Now , so if we use the action of on before transferring to , we get . Be careful not to confuse the counit with the basis elements .

On the other hand, if we transfer first, we must calculate

Now let’s use the fact that we’ve got this basis sitting around to expand out both and as matrices. We’ll just take on matrix indices on the right for our notation. Then we continue the calculation above:

And so the coevaluation map does indeed intertwine the two actions of . Together with the evaluation map, it provides the duality on the category of representations of a Hopf algebra that we were looking for.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.